Tsallis Relative Entropy and Anomalous Diffusion

Abstract

In this paper we utilize the Tsallis relative entropy, a generalization of the Kullback-Leibler entropy in the frame work of non-extensive thermodynamics to analyze the properties of anomalous diffusion processes. Anomalous (super-) diffusive behavior can be described by fractional diffusion equations, where the second order space derivative is extended to fractional order α ∈ (1, 2). They represent a bridging regime, where for α = 2 one obtains the diffusion equation and for α = 1 the (half) wave equation is given. These fractional diffusion equations are solved by so-called stable distributions, which exhibit heavy tails and skewness. In contrast to the Shannon or Tsallis entropy of these distributions, the Kullback and Tsallis relative entropy, relative to the pure diffusion case, induce a natural ordering of the stable distributions consistent with the ordering implied by the pure diffusion and wave limits.

An Entropy-Based Network Anomaly Detection Method

“…The Shannon entropy assumes a tradeoff between contributions from the main mass of the distribution and the tail. With the parameterized Tsallis [76][77][78] or Renyi [79,80] entropy, one can control this tradeoff. In general, if the parameter denoted as α has a positive value, it exposes the main mass, if the value is negative -it refers to the tail.…”

Time Evolution of Relative Entropies for Anomalous Diffusion

“…The space-fractional diffusion equation represents a family of processes in the bridge regime that can be ordered by the parameter α, which will be called the bridge ordering. In [43], the relative entropies (e.g., Kullback-Leibler), in contrast to the regular entropies, order the PDF's from Equation (1), because of a monotonic relationship in α, placing the wave and diffusion limits "farthest" from each other, even if not in a metrical sense. This establishes relative entropies as a natural measure for the bridging regime.…”

“…This establishes relative entropies as a natural measure for the bridging regime. However, [43] considered circumstances at one particular time. This paper extends the previous work by asking whether this ordering is preserved over all time.…”

Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method

“…For example, the fluid-dynamic traffic model with fractional derivatives can eliminate the deficiency arising from the assumptions of continuum traffic flow [9]. It is of interest to note that the fractional calculus theory and its application have been studied in great detail in the literature ( [10][11][12][13][14] and the references therein).…”